In 2011, independently in [1]} and [2]}, it was established a general theory to study matrix-valued orthogonal polynomials through Riemann–Hilbert problems, thus extending the well known result of Fokas, Its and Kitaev [3]} from the scalar to the matrix case. This theory, in our opinion, provides one of the best ways to deduce, in a uniform way, non-linear equations related to matrix-valued orthogonal polynomials, using the well known technique of reducing the Riemann–Hilbert problem to a simpler one, where the jumps are constant. In [1]}, this technique has been used to deduce a matrix version of the discrete Painlevé I equation, while in [5]} we proved that the Christoffel–Darboux kernel associated to certain matrix-valued Hermite polynomials satisfies a non-commutative version of the Painlevé IV equation.
